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Mirrors > Home > MPE Home > Th. List > dfiun3 | Structured version Visualization version GIF version |
Description: Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfiun3.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
dfiun3 | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiun3g 5828 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
2 | dfiun3.1 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ V) |
4 | 1, 3 | mprg 3151 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∪ cuni 4831 ∪ ciun 4912 ↦ cmpt 5139 ran crn 5549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: tgrest 21762 comppfsc 22135 sigapildsys 31442 ldgenpisyslem1 31443 dstfrvunirn 31753 ctbssinf 34711 mblfinlem2 34965 volsupnfl 34972 istotbnd3 35082 sstotbnd 35086 fourierdlem80 42545 |
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